A353694 -id:A353694 - cp's OEIS frontend

A353693 a(n) is the least multiplier k such that the exponents in the prime factorization of k*n are mutually distinct (A130091).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 3, 2, 5, 2, 1, 2, 3, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 2, 1, 6, 1, 2, 1, 1, 5, 12, 1, 1, 3, 20, 1, 1, 1, 2, 1, 1, 7, 12, 1, 1, 1, 2, 1, 6, 5, 2

Offset: 1

Amiram Eldar, May 04 2022

nonn

First differs from A327499 at n = 30.
If n = Product_{i=1..k} p_i is squarefree (A005117), and p_1 < p_2 < ... < p_k are its k ordered prime divisors, then a(n) = Product_{i} p_i^(k-i).
If n is powerful (A001694) then a(n) = a(n/rad(n)), where rad(n) is the squarefree kernel of n (A007947). In general, if k = A051904(n) is the minimal exponent in the prime factorization of n, then a(n) = a(n/(rad(n)^(k-1))).

			a(2) = 1 since 2 = 2^1 has only one exponent (1) in its prime factorization.
a(6) = 2 since 6 = 2*3 has two equal exponents (1) in its prime factorization, and 2*6 = 12 = 2^2*3 has two distinct exponents (1 and 2).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A005117, A007947, A130091, A130092, A327498, A327499, A353694.

a[n_] := Module[{k = 1}, While[!UnsameQ @@ FactorInteger[k*n][[;; , 2]], k++]; k]; Array[a, 100]

a(n) = my(k=1, f=factor(n)[,2]); while(#Set(f) != #f, k++; f=factor(k*n)[,2]); k; \\ Michel Marcus, May 05 2022

a(n) = 1 if and only if n is in A130091.
a(A130092(n)) > 1.
rad(a(n)) | rad(n).
a(n) = A353694(n)/n.

cp's OEIS Frontend

A353693 a(n) is the least multiplier k such that the exponents in the prime factorization of k*n are mutually distinct (A130091).

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